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Instead of working directly at the level of vector spaces, we take a step back and first study the decomposition problem at the level of sets. This approach allows us to define the combinatorial notion of\n                    <jats:italic>rooted subsets<\/jats:italic>\n                    . In the case of a filtered metric space\n                    <jats:italic>M<\/jats:italic>\n                    , rooted subsets relate the clustering behavior of the points of\n                    <jats:italic>M<\/jats:italic>\n                    with the decomposition of the associated persistence module. In particular, we can identify intervals in such a decomposition quickly. In addition, rooted subsets can be understood as a generalization of the elder rule, and are also related to the notion of constant conqueror of Cai, Kim, M\u00e9moli and Wang. As an application, we give a lower bound on the number of intervals that we can expect in the decomposition of zero-dimensional persistence modules of a density-Rips filtration in Euclidean space: in the limit, and under very general circumstances, we can expect that at least 25% of the indecomposable summands are interval modules.\n                  <\/jats:p>","DOI":"10.1007\/s00454-024-00700-7","type":"journal-article","created":{"date-parts":[[2024,11,16]],"date-time":"2024-11-16T01:42:20Z","timestamp":1731721340000},"page":"818-838","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets"],"prefix":"10.1007","volume":"74","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5822-546X","authenticated-orcid":false,"given":"\u00c1ngel Javier","family":"Alonso","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael","family":"Kerber","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,11,16]]},"reference":[{"key":"700_CR1","doi-asserted-by":"publisher","unstructured":"Asashiba, H., Buchet, M., Escolar, E.G., Nakashima, K., Yoshiwaki, M.: On interval decomposability of 2D persistence modules. 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